Splitting-Off in Hypergraphs

Authors Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király , Shubhang Kulkarni



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Kristóf Bérczi
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, IL, USA
Tamás Király
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
Shubhang Kulkarni
  • University of Illinois, Urbana-Champaign, IL, USA

Acknowledgements

Part of this work was done while Karthekeyan and Shubhang were visiting Eötvös Loránd University. Karthekeyan thanks Eklavya Sharma for engaging in preliminary discussions on hypergraph splitting-off.

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Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni. Splitting-Off in Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.23

Abstract

The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász [Lov{á}sz, 1974; Lov{á}sz, 1993] and Mader [Mader, 1978] showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature [Lovász, 1976; Mader, 1978; Frank, 1993; Frank and Király, 2002; Király and Lau, 2008; Frank, 1992; Goemans and Bertsimas, 1993; Frank, 1994; Bang-Jensen et al., 1995; Frank, 2011; Nagamochi and Ibaraki, 2008; Nagamochi et al., 1997; Henzinger and Williamson, 1996; Goemans, 2001; Jordán, 2003; Kriesell, 2003; Jain et al., 2003; Chan et al., 2011; Bhalgat et al., 2008; Lau, 2007; Chekuri and Shepherd, 2008; Nägele and Zenklusen, 2020; Blauth and Nägele, 2023]. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of k-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau [Király and Lau, 2008]). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network optimization
  • Theory of computation → Routing and network design problems
Keywords
  • Hypergraphs
  • Hypergraph Connectivity
  • Splitting-off
  • Constructive Characterizations
  • Hypergraph Orientations
  • Submodular Functions
  • Combinatorial Optimization

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